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Wall-pressure and velocity statistics in the turbulent boundary layer (TBL) on a cambered controlled-diffusion aerofoil at $8^{\circ }$ incidence, a Mach number of 0.25 and a chord-based Reynolds number ${Re}_c=1.5\times 10^{5}$ are analysed at four locations on the suction side with zero and adverse pressure gradients (ZPG and APG), characterised by increasing Reynolds numbers based on momentum thickness, ${Re}_{\theta }=319$ , 390, 877 and $1036$ . The strong APG yields a highly non-equilibrium TBL at the trailing edge that significantly affects the turbulent flow statistics. Different normalisations of the full wall-pressure statistics involved in trailing-edge noise are analysed for the first time in such strong APG with convex curvature, and compared with available experimental and numerical data. Good overall agreement is found in the ZPG region, and most results obtained in previous APG TBL can be extended to the present highly non-equilibrium case. The presence of strong APG augments the intensity of wall-pressure fluctuations noticeably at low frequencies, shortens the streamwise and broadens the spanwise coherence of wall-pressure fluctuations in both time and space, and significantly reduces the convection velocity. The wall-pressure power spectral density are found to scale with the displacement thickness, the Zaragola–Smits velocity and the root-mean-squared pressure, the latter possibly being replaced by the local maximum Reynolds shear stress. The other two key parameters to trailing-edge noise modelling, the spanwise coherence length and the convection velocity, rather scale with displacement thickness and friction velocity, respectively.

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... They showed that none of the outer, inner or mixed scaling collapsed the wall pressure spectra in all regions of the flow. Normalization with the local maximum magnitude of the Reynolds shear stress, however, was shown to collapse the low-frequency range of WPS for APG flows including those with separation (Abe 2017;Ji & Wang 2012;Caiazzo et al. 2023). ...

... As the separation point is approached, R t tends to 0 and β to infinity. This indicates issues in many existing WPS models when used for strong-APG flows near incipient separation (Caiazzo et al. 2023), which are examined in detail in Section 5. (2000), which were carried out for ZPG flows only but reached higher Reynolds numbers. ...

... These observations indicate that wall-pressure r.m.s. scales better with ρ|u ′ v ′ | max than with q e or τ w , in attached flows under strong pressure gradients; similar observations were made by Na & Moin (1998), Abe (2017) and Caiazzo et al. (2023). However, the appropriate wall-pressure as shown by Caiazzo et al. (2023)). ...

This study uses high-fidelity simulations (DNS or LES) and experimental datasets to analyse the effect of non-equilibrium streamwise mean pressure gradients (adverse or favourable), including attached and separated flows, on the statistics of boundary layer wall-pressure fluctuations. The datasets collected span a wide range of Reynolds numbers (Re θ from 300 to 23,400) and pressure gradients (Clauser parameter from −0.5 to 200). The datasets are used to identify an optimal set of variables to scale the wall pressure spectrum: edge velocity, boundary layer thickness, and the peak magnitude of Reynolds shear stress. Using the present datasets, existing semi-empirical models of wall-pressure spectrum are shown unable to capture effects of strong, non-equilibrium adverse pressure gradients, due to inappropriate scaling of wall pressure using wall shear stress, calibration with limited types of flows, and dependency on model parameters based on friction velocity, which reduces to zero at the detachment point. To address these shortcomings , a generalized wall-pressure spectral model is developed with parameters that characterize the extent of the logarithmic layer and the strength of the wake. Derived from the local mean velocity profile, these two parameters inherently carry effect of the Reynolds number, as well as those of the non-equilibrium pressure gradient and its history. Comparison with existing models shows that the proposed model behaves well and is more accurate in strong-pressure-gradient flows and in separated-flow regions.

... The velocity profile at RMP 26 (x/C = 0.98) shows when the CD airfoil placed at 15 • and 16 m/s flow incidence and inlet velocity respectively, the near wall mean velocity is reduced compared to the inlet velocity. Similar observations have been made by Caiazzo et al. [2023] (see figure 4), who reported a decrease in the near wall mean velocity as the mean pressure gradient increases. As such, we expect the boundary layer to grow faster in the streamwise direction near the trailing-edge region thickness (Re θ ) for the airfoil placed at 15 • angle-of-attack is substantially higher than that of the 8 • case near the trailing-edge region. ...

... This normalization holds true because, as first shown by Na and Moin [1998], the term p rms /(−ρ u 1 u 2max ) falls between 2 and 3 for boundary-layer flows. This was later confirmed by [Abe 2017;Le Floc'h et al. 2020] for canonical boundarylayer flows, and more recently by Caiazzo et al. [2023] for flows past an airfoil. These observations are confirmed in tables III and IV for the present case. ...

... These observations are confirmed in tables III and IV for the present case. Small deviations from the aforementioned values can be ascribed to measurement uncertainty, and the presence of open jet, which predominantly contributes to low frequency wall-pressure spectra and which is absent in the aforementioned data [Abe 2017;Caiazzo et al. 2023;Le Floc'h et al. 2020;Pargal 2023]. More importantly, when the scaling proposed by Pargal [2023] is used to scale the wall-pressure spectra in Figure 15, a collapse in the low-frequency range is achieved. ...

This paper presents a detailed aeroacoustic investigation of a controlled-diffusion airfoil at near-stall condition. The study aims at answering two research questions: identify the flow mechanism responsible for separation noise for an airfoil near-stall conditions and whether the noise is generated by a dipole for airfoil close to stall and can be quantified by Amiet's diffraction theory. The study uses synchronized particle image velocimetry, remote microphone probes, and far-field microphone measurements to perform experiments at two chord-based Reynolds numbers of about 150 000 and 250 000. The results show that when the airfoil is placed at a higher angle of attack, such as 15°, strong amplification of flow disturbance is seen, resulting in the rolling up of the shear layer in the aft region of the airfoil, forming large coherent structures. While these rollers play a central role in the increase in noise due to flow separation, the flapping of shear layer does not contribute to the separation noise. The present study conclusively shows that separation noise is dipolar in nature and that the quadrupolar contribution for low-speed airfoils at near-stall conditions can be neglected. However, the increase in flow disturbances measured close to the trailing edge of the airfoil implies that the assumption of small-amplitude disturbance is no longer valid, which is the central premise of the thin linearized airfoil theory. Outside the frequency range at which flow separation operates, Amiet's theory is able to predict the far-field noise even at high angles of attack.

... The velocity profile at RMP 26 (x/C = 0.98) shows when the CD airfoil placed at 15 • and 16 m/s flow incidence and inlet velocity respectively, the near wall mean velocity is reduced compared to the inlet velocity. Similar observations have been made by Caiazzo et al. [2023] (see figure 4), who reported a decrease in the near wall mean velocity as the mean pressure gradient increases. As such, we expect the boundary layer to grow faster in the streamwise direction near the trailing-edge region thickness (Re θ ) for the airfoil placed at 15 • angle-of-attack is substantially higher than that of the 8 • case near the trailing-edge region. ...

... This normalization holds true because, as first shown by Na and Moin [1998], the term p rms /(−ρ u 1 u 2max ) falls between 2 and 3 for boundary-layer flows. This was later confirmed by [Abe 2017;Le Floc'h et al. 2020] for canonical boundarylayer flows, and more recently by Caiazzo et al. [2023] for flows past an airfoil. These observations are confirmed in tables III and IV for the present case. ...

... These observations are confirmed in tables III and IV for the present case. Small deviations from the aforementioned values can be ascribed to measurement uncertainty, and the presence of open jet, which predominantly contributes to low frequency wall-pressure spectra and which is absent in the aforementioned data [Abe 2017;Caiazzo et al. 2023;Le Floc'h et al. 2020;Pargal 2023]. More importantly, when the scaling proposed by Pargal [2023] is used to scale the wall-pressure spectra in Figure 15, a collapse in the low-frequency range is achieved. ...

This paper presents a detailed aeroacoustic investigation of a Controlled-Diffusion airfoil at near stall condition. The study aims at answering two research questions: identify the flow mechanism responsible for separation noise for an airfoil near stall conditions and whether the noise is generated by a dipole for airfoil close to stall and can be quantified by Amiet's diffraction theory. The study uses synchronized PIV, RMP and far-field microphone measurements to perform experiments at two chord based Reynolds numbers of about 150,000 and 250,000. The results show that when the airfoil is placed at a higher angle of attack, such as $15^{\circ}$, strong amplification of flow disturbance is seen, resulting in the rolling up of the shear layer in the aft-region of the airfoil, forming large coherent structures. While these rollers play a central role in the increase in noise due to flow separation, the flapping of shear layer does not contribute to the separation noise. The present study conclusively shows that separation noise is dipolar in nature, and that the quadrupolar contribution for low-speed airfoils at near-stall conditions can be neglected. However, the increase in flow disturbances measured close to the trailing-edge of the airfoil implies that the assumption of small amplitude disturbance is no longer valid, which is the central premise of the thin linearized airfoil theory. Outside the frequency range at which flow separation operates, Amiet's theory is able to predict the far-field noise even at high angles of attack.

... The right terms are associated to the noise sources and correspond to the wall-pressure spectra ( ) and the spanwise coherence length 2 ( ). The latter is modelled using a Corcos model in first approximation [22]. The former is obtained from empirical wall-pressure spectra. ...

... A parallel effort has been to develop a generalized semi-empirical model for non-equilibrium boundary layers with strong pressure gradients (APG/FPG) and regions with separation (typical in highly-loaded low-speed fans) and reattachment. Based on previous findings on the scaling of wall pressure r.m.s [19,22,26,27], Pargal [23] chose the local maximum Reynolds shear stress magnitude, | ′ ′ | max , as the pressure scale instead of that based on , which was used in Goody, Rozenberg or Lee models. Indeed, it was pointed out that neither nor are appropriate parameters for separating TBLs as they tend to 0 and ∞, respectively, close to the separation point [23]. ...

... An analysis of the coherence of the wall-pressure statistics is then carried out to assess the reliability of the DNS computations. The coherence is defined as follows [28]: ...

Direct numerical simulations (DNSs) of an incompressible turbulent boundary layer on an airfoil (suction side) and that on a flat plate are compared to characterize the non-equilibrium turbulence and the effect of wall curvature on the flow. The two simulations effectively impose matching streamwise distributions of adverse pressure gradient (APG) quantified by the acceleration parameter (K). For the airfoil flow, an existing compressible DNS carried out by Wu et al. ["Effects of pressure gradient on the evolution of velocity-gradient tensor invariant dynamics on a controlled-diffusion aerofoil at Re c ¼ 150,000," J. Fluid Mech. 868, 584-610 (2019)] of the flow around a controlled-diffusion airfoil is used. For the flat-plate flow, a separate simulation is carried out with the aim to reproduce the flow in the region of the airfoil boundary layer with zero to adverse pressure gradients. Comparison between the two cases extracts the effect of a mild convex wall curvature on velocity and wall-pressure statistics in the presence of APG. In the majority part of the boundary layer development, curvature effect on the flow is masked by that of the APG, except for the region with weak pressure gradients or a thick boundary layer where the effect of wall curvature appears to interact with that of APG. High-frequency wall-pressure fluctuations are also augmented by the wall curvature. Overall, the boundary layers are qualitatively similar with and without the wall curvature. This indicates that a flat-plate boundary layer DNS may serve as a low-cost surrogate of a boundary layer over the airfoil or other objects with mild curvatures to capture important flow features to aid modeling efforts.

When hydrodynamic energy within a turbulent boundary layer is scattered by a sharp trailing edge, the hydrodynamic energy is converted to acoustic energy, which propagates to the far field. This trailing-edge noise occurs in aircraft wing, turbomachinery blades, wind turbine blades, helicopter blades, etc. Being dominant at high frequencies, this trailing-edge noise is a key element that annoys human hearing. This article covers virtually the entire landscape of modern research into trailing-edge noise including theoretical developments, numerical simulations, wind tunnel experiments, and applications of trailing-edge noise. The theoretical approach includes Green's function formulations, Wiener-Hopf methods that solve the mixed boundary-value problem, Howe's and Amiet's models that relate the wall pressure spectrum to acoustic radiation. Recent analytical developments for poroelasticity and serrations are also included. We discuss a hierarchy of numerical approaches that range from semi-empirical schemes that estimate the wall pressure spectrum using mean-flow and turbulence statistics to high-fidelity unsteady flow simulations such as Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS) that resolve the sound generation and scattering process based on the first-principles flow physics. Wind tunnel experimental research that provided benchmark data for numerical simulations and unravel flow physics is reviewed. In each theoretical, numerical, and experimental approach, noise control methods for mitigating trailing-edge noise are discussed. Finally, highlights of practical applications of trailing-edge noise prediction and reduction to wind turbine noise, fan noise, and rotorcraft noise are given. The current challenges in each approach are summarized with a look toward the future developments. The review could be useful as a primer for new researchers or as a reference point to the state of the art for experienced professionals.

While the computation of the boundary-layer thickness is straightforward for canonical equilibrium flows, there are no established definitions for general non-equilibrium flows. In this work, a new method is developed based on a local reconstruction of the "inviscid" velocity profile UI [y] resulting from the Bernoulli equation. The boundary-layer thickness δ99 is then defined as the location where U/UI = 0.99, which is consistent with its classical definition for the zero-pressure-gradient boundary layers (ZPGBLs). The new method is parameter free, and can be deployed for both internal and external flows without resorting to an iterative procedure, numerical integration, or numerical differentiation. The superior performance of the new method over various existing methods is demonstrated by applying the methods to laminar and turbulent boundary layers and two flows over airfoils. Numerical experiments reveal that the new method is more accurate and more robust than existing methods, and it is applicable for flows over a wide range of Reynolds numbers.

Two-point velocity statistics near the trailing edge of a controlled diffusion airfoil are obtained, both experimentally and analytically, by decomposing Poisson’s equation for pressure into the mean-shear (MS) and turbulence–turbulence (TT) interaction terms. The study focuses on the modeling of each interaction term, in order to allow for the reconstruction of the wall-pressure spectra from tomographic velocimetry data, without numerically solving for pressure. The two-point correlation of the wall-normal velocity that describes the magnitude of the MS source term is found to be influenced by various competing factors such as blocking, mean-shear, and the adverse mean pressure gradient. The blocking term is found to supersede the other interaction terms close to the wall, making the two-point velocity correlation self-similar. The most dominant TT term that contributes to far-field noise for an observer located perpendicular to the airfoil chord at the mid-span is shown to be the one that quantifies the variation of the wall-normal velocity fluctuations in the longitudinal direction because of the statistical homogeneity of turbulence in planes parallel to the wall. A model to determine the contribution of the TT interaction term is proposed where the fourth-order two-point correlation can be modeled using Lighthill’s approximation. However, its contribution toward wall-pressure spectra is found to be substantially lower than the MS term in the present case.

Non-equilibrium development in turbulent boundary layers with changing pressure gradients - Volume 897 - Ralph J. Volino

The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at 0 • and 5 • angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length Re c = 400, 000. The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter β (Vinuesa et al., Flow Turbul. Combust., vol. 99, 2017, pp. 565-587), through the study of cases with matching local values of β and the friction Reynolds number Re τ to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region (Baars et al., J. Fluid Mech., vol. 823, 2017, R2), unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors' knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

This paper presents a comprehensive analytical approach to the modelling of wall pressure fluctuations under a turbulent boundary layer, unifying and expanding the analytical models that have been proposed over many decades. The Poisson equation governing pressure fluctuations is Fourier-transformed in the wavenumber domain to obtain a modified Helmholtz equation, which is solved with a Green's function technique. The source term of the differential equations is composed of turbulence-mean shear and turbulence-turbulence interaction terms, which are modelled separately within the
hypothesis of joint normal probability distribution of the turbulent field. The functional expression of turbulence statistics is shown to be the most critical point for a correct representation of the wall-pressure spectrum. The effect of various assumptions on the shape of the longitudinal correlation function of turbulence is assessed in the first place with purely analytical considerations using an idealised flow model. Then, the effect of the hypothesis on the spectral distribution of boundary-layer turbulence on the resulting wall-pressure spectrum is compared with the results of DNS computations and pressure
measurements on a controlled-diffusion aerofoil. The boundary layer developing over the suction side of this aerofoil in test conditions is characterized by an adverse pressure gradient. The final part of the paper discusses the numerical aspect of wall-pressure spectrum computation. A Monte-Carlo technique is used for a fast evaluation of the multi-dimensional integral formulation developed in the theoretical part.

This paper presents a theoretical study of active control of turbulent boundary layer TBL induced sound transmission through the cavity-backed double panels. The aerodynamic model used is based on the Corcos wall pressure distribution. The structural-acoustic model encompasses a source panel (skin panel), coupled through an acoustic cavity to the radiating panel (trim panel). The radiating panel is backed by a larger acoustic enclosure (the back cavity). A feedback control unit is located inside the acoustic cavity between the two panels. It consists of a control force actuator and a sensor mounted at the actuator footprint on the radiating panel. The control actuator can react off the source panel. It is driven by an amplified velocity signal measured by the sensor. A fully coupled analytical structural-acoustic model is developed to study the effects of the active control on the sound transmission into the back cavity. The stability and performance of the active control system are firstly studied on a reduced order model. In the reduced order model only two fundamental modes of the fully coupled system are assumed. Secondly, a full order model is considered with a number of modes large enough to yield accurate simulation results up to 1000 Hz. It is shown that convincing reductions of the TBL-induced vibrations of the radiating panel and the sound pressure inside the back cavity can be expected. The reductions are more pronounced for a certain class of systems, which is characterised by the fundamental natural frequency of the skin panel larger than the fundamental natural frequency of the trim panel.

This study investigates the effects of a pressure gradient on the wall pressure beneath equilibrium turbulent boundary layers. Excitation of the walls of a vehicle by turbulent boundary layers indeed constitutes a major source of interior noise and it is necessary to take into account the presence of a pressure gradient to represent the effect of the curvature of the walls. With this aim, large-eddy simulations of turbulent boundary layers in the presence of both mild adverse and mild favourable pressure gradients are carried out by solving the compressible Navier–Stokes equations. This method provides both the aeroacoustic contribution and the hydrodynamic wall-pressure fluctuations. A critical comparison with existing databases, including recent measurements, is conducted to assess the influence of a free stream pressure gradient. The analyses of wall-pressure spectral densities show an increase in the low-frequency content from adverse to favourable conditions, yielding higher integrated levels of pressure fluctuations scaled by the wall shear stress. This is accompanied by a steeper decay rate in the medium-frequency portion for adverse pressure gradients. No significant difference is found for the mean convection velocity. Frequency–wavenumber spectra including the subconvective region are presented for the first time in the presence of a pressure gradient. A scaling law for the convective ridge is proposed, and the acoustic domain is captured by the simulations. Direct acoustic emissions have similar features in all gradient cases, even if slightly higher levels are noted for boundary layers subjected to an adverse gradient.

This manuscripts presents a study on adverse-pressure-gradient turbulent boundary layers under different Reynolds-number and pressure-gradient conditions. In this work we performed Particle Image Velocimetry (PIV) measurements supplemented with Large-Eddy Simulations in order to have a dataset covering a range of displacement-thickness-based Reynolds number
2300< Reds <34000 and values of the Clauser pressure-gradient parameter beta� up to 2.4. The spatial resolution limits of PIV for the estimation of turbulence statistics have been overcome via ensemble-based approaches. A comparison between ensemble-correlation and ensemble Particle Tracking Velocimetry was carried out to assess the uncertainty of the two methods. The effects of beta�, Re and of the pressure-gradient history on turbulence statistics were assessed. A modal analysis via Proper Orthogonal Decomposition was carried out on the flow �elds and showed that about 20% of the energy contribution corresponds to the fi�rst mode, while 40% of the turbulent kinetic energy corresponds to the fi�rst four modes with no appreciable dependence on � beta and Re within the investigated range. The topology of the spatial modes shows a dependence on the Reynolds number and on the pressure-gradient strength, in line with the results obtained from the analysis of the turbulence statistics. The contribution of the modes to the Reynolds stresses and the turbulence production was assessed using a truncated low-order reconstruction with progressively larger number of modes. It is shown that the outer peaks in the Reynolds-stress pro�files are mostly due to large-scale structures in the outer part of the boundary layer.

A direct numerical simulation database of the flow around a NACA4412 wing section at Rec = 400,000 and 5 degree angle of attack (Hosseini et al., Int. J. Heat Fluid Flow, vol. 61, pp. 117-128), obtained with the spectral-element code Nek5000, is analyzed. The Clauser pressure-gradient parameter beta� ranges from 0 and 85 on the suction side, and from 0 to -0.25 on the pressure side of the wing. The maximum Re_tau and Re_theta values are around 2,800 and 373 on the suction side, respectively, whereas on the pressure side these values are 818 and 346. Comparisons between the suction side with zero-pressure-gradient turbulent boundary layer data show larger values of the shape factor and a lower skin friction, both connected with the fact that the adverse pressure
gradient present on the suction side of the wing increases the wall-normal convection. The adverse-pressure-gradient boundary layer also exhibits a more prominent wake region, the development of an outer peak in the Reynolds-stress tensor components, and increased production and dissipation across the boundary layer. All these effects are connected with the fact that the large-scale motions of the flow become relatively more intense due to the adverse pressure gradient, as apparent from spanwise premultiplied power-spectral density maps. The emergence of an outer spectral peak is observed at � values of around 4 for lambda_z = 0:65*delta_99, closer to the wall than the spectral outer peak observed in zero-pressure-gradient turbulent boundary layers at higher Re_theta. The effect of the slight favorable pressure gradient present on the pressure side of the wing is opposite the one of the adverse pressure gradient, leading to less energetic outer-layer structures.

Significance
Uncovering the constitutive coherent structure in the inner layer of the canonical turbulent boundary layer has remained a central fluid mechanics theme, because it tests our intellectual ability to understand even the simplest external flow. We describe here how turbulent spots are initiated in bypass boundary-layer transition and uncover the ubiquity of concentrations of vortices in the fully turbulent region with characteristics remarkably like transitional–turbulent spots. We present strong evidence that these concentrations of vortices are the constitutive coherent structure of the inner layer near the wall. This study contributes to the unification of understanding of phenomena occurring in boundary-layer late-stage transition with near-wall turbulent boundary-layer structure and dynamics in the developed flow.

Turbulent boundary layers under adverse pressure gradients are studied using well-resolved large-eddy simulations (LESs) with the goal of assessing the influence of the streamwise pressure-gradient development. Near-equilibrium boundary layers were characterized through the Clauser pressure-gradient parameter β. In order to fulfil the near-equilibrium conditions, the freestream velocity was prescribed such that it followed a power-law distribution. The turbulence statistics pertaining to cases with a constant value of β (extending up to approximately 40 boundary-layer thicknesses) were compared with cases with non-constant β distributions at matched values of β and friction Reynolds number Re τ. An additional case at matched Reynolds number based on displacement thickness Re δ * was also considered. It was noticed that non-constant β cases appear to approach the conditions of equivalent constant β cases after long streamwise distances (around 7 boundary-layer thicknesses). The relevance of the constant β cases lies in the fact that they define a " canonical " state of the boundary layer, uniquely char-acterised by β and Re. The investigations on the flat plate were extended to the flow around a wing section overlapping in terms of β and Re. Comparisons with the flat-plate cases at matched values of β and Re revealed that the different development history of the turbulent boundary layer on the wing section leads to a less pronounced wake in the mean velocity as well as a weaker second peak in the Reynolds stresses. This is due to the weaker accumulated effect of the β history. Furthermore, a scaling law suggested by Kitsios et al. (Int. J. Heat Fluid Flow, 2016), proposing the edge velocity and the displacement thickness as scaling parameters, was tested on two constant-pressure-gradient parameter cases. The mean velocity and Reynolds-stress profiles were found to be dependent on the downstream development. The present work is the first step towards assessing history effects in adverse-pressure-gradient turbulent boundary layers and highlights the fact that the values of the Clauser pressure-gradient parameter and the Reynolds number are not sufficient to characterise the state of the boundary layer.

This study presents the first set of experiments with the Controlled-Diffusion (CD) airfoil in the newly-built anechoic wind-tunnel at Université de Sherbrooke. Velocity measurements in the latter show very uniform mean flow and low turbulence level (0.4 %) up to 56 m/s in the 30 cm square nozzle exit section. Acoustic and velocity measurements have been carried out at several flow velocities and angles of attack. Three distinct flow regimes are observed. At high angle of attack and high velocity the usual broadband noise signature found in the Ecole Centrale de Lyon anechoic wind tunnels is recovered. At low angle of attack, the power spectral density of the microphone signal is dominated by a primary tone with secondary tones, typical of Tollmien-Schlichting noise radiation. This dominant tone is also recovered in the power spectral density of the flow velocity signal. Signal processing tools (spectrogram and bicoherence) are used to investigate the presence of the secondary tones. Two tonal regimes can then be distinguished: one stationnary and one intermittent where some tones disappear intermittently and are formed by a non-linear process. Finally, two parallel microphone arrays associated with high resolution imaging is used to localize the noise sources at the primary tone frequency, which are found at the airfoil trailing edge. At this low frequency, the L1-GIB algorithm is clearly seen more efficient than classical beamforming.

Two-point statistics are presented for a new direct simulation of the zero-pressure- gradient turbulent boundary layer in the range Reθ = 2780–6680, and compared with channels in the same range of Reynolds numbers, δ+ ≈ 1000–2000. Three- dimensional spatial correlations are investigated in very long domains to educe the average structure of the velocity and pressure fluctuations. The streamwise velocity component is found to be coherent over longer distances in channels than in bound- ary layers, especially in the direction of the flow. For weakly correlated structures, the maximum streamwise length is O(7δ) for boundary layers and O(18δ) for chan- nels, attained at the logarithmic and outer regions, respectively. The corresponding lengths for the spanwise and wall-normal velocities and for the pressure are shorter, O(δ-2δ). The correlations are shown to be inclined to the wall at angles that depend on the distance from the wall, on the variable being considered, and on the correlation level used to define them. All these features change little between the two types of flows. Most the above features are also approximately independent of the Reynolds number, except for the pressure, and for the streamwise velocity structures in the channel. Further insight into the flow is provided by correlations conditioned on the intensity of the perturbations at the reference point, or on their sign. The statistics of the new simulation are available in our website.

The goal of this experimental study is to investigate wall pressure wavenumber-frequency spectra induced by a turbulent boundary layer in the presence of a mean pressure gradient. The mean pressure gradient is achieved by changing the ceiling height of a rectangular channel flow. Wall pressure spectra are measured for zero-, adverse- and favorable-pressure-gradient boundary layers by using a pinhole microphone combined to a high-frequency calibration of the sensor. A linear antenna based on a non-uniform distribution of remote microphones mounted on a rotating disk is also developed to obtain a direct measurement of both aerodynamic and acoustic components of wavenumber-frequency spectra. First results, comparisons and analyses are then discussed.

An empirical model to predict the wall-pressure fluctuation spectra beneath adverse pressure gradient flows is presented. It is based on Goody's model, which already incorporates the effect of Reynolds number but is limited to zero pressure gradient flows. The extension relies on six test cases from five experimental or numerical studies covering a large range of Reynolds number, 5.6 × 10 2 < R θ < 1.72 × 10 4, in both internal (channel) and external (airfoil) flows. A review of the boundary-layer parameters characterizing the pressure gradient effects is provided, and the more relevant ones are introduced as new variables in the model. The method is then compared to the zero pressure gradient model it is derived from. The influence of the pressure gradient on the wall-pressure spectrum is discussed. Finally, the method is applied to provide input data of a radiated trailing-edge noise model by means of an aeroacoustic analogy, namely Amiet's theory of turbulent boundary layer past a trailing edge. The results are compared to experimental data obtained in an open-jet anechoic wind tunnel. Copyright © 2012 by Yannick Rozenberg, Gilles Robert, and Stéphane Moreau.

We analyze and compare various empirical models of wall pressure spectra beneath turbulent boundary layers and propose an alternative machine learning approach using Artificial Neural Networks (ANNs). The analysis and the training of the ANN are performed on data from experiments and high-fidelity simulations by various authors, covering a wide range of flow conditions. We present a methodology to extract all the turbulent boundary layer parameters required by these models, also considering flows experiencing strong adverse pressure gradients. Moreover, the database is explored to unveil important dependencies within the boundary layer parameters and to propose a possible set of features from which the ANN should predict the wall pressure spectra. The results show that the ANN outperforms traditional models in adverse pressure gradients, and its predictive capabilities generalize better over the range of investigated conditions. The analysis is completed with a deep ensemble approach for quantifying the uncertainties in the model prediction and integrated gradient analysis of the model sensitivity to its inputs. Uncertainties and sensitivities allow for identifying the regions where new training data would be most beneficial to the model's accuracy, thus opening the path toward a self-calibrating modeling approach.

The wall pressure fluctuations induced by turbulent boundary layers on a flat plate model were measured with an L-shaped array of Kulite miniature pressure sensors. By installing a NACA airfoil above the plate with an adjustable angle of attack, different adverse pressure gradients were produced on the plate. The streamwise and spanwise coherence and coherence lengths of the wall pressure fluctuations are calculated for zero and adverse pressure gradient flows. The effect of the pressure gradient on the coherence and coherence lengths in both streamwise and spanwise directions is quantified. Based on the results of the present test cases and selected published datasets in zero pressure gradients, covering the range of Reynolds numbers based on the momentum thickness between 3300 and 42000, the Reynolds number dependence of the streamwise wall pressure coherence is discussed and mathematically expressed. A coherence length model for zero pressure gradient boundary layers, taking into account the Reynolds number effect, is proposed based on scaled spectra of the coherence lengths. In comparison with the other published models, the present model achieves a considerable improvement of the prediction accuracy for boundary layer flows, covering a large range of Reynolds numbers. Furthermore, an evaluation of the Corcos model and the Smol’yakov and Tkachenko model for prediction of the off-axis coherence of the fluctuating wall pressure field is made.

A compressible direct numerical simulation of the flow and near-field acoustics
over a Controlled-Diffusion aerofoil at 8 degree geometrical angle of attack, a Reynolds
number of Rec = 150000 based on the chord and a free-stream Mach number
of M = 0.25, with a spanwise extent of 12% chord was conducted to produce
a space-time database to determine the aerofoil noise generation mechanisms.
Three main noise sources respectively from the flow separation and reattachment
at the leading-edge on the suction side, the interaction between the attached
turbulent flow on the suction side and the trailing-edge and an additional
near-wake source due to secondary instability are observed. It should be noticed, however, that such an additional source may strongly depend on the aerofoil shape and also on the flow conditions. The contribution of these three noise sources to the noise radiation at different frequencies is analysed through a wavenumber-frequency filtering procedure. In the low to mid frequency range, the contribution of the traditional trailing-edge noise is significant, whereas at high frequencies, the noise radiation is shown to be related to the flow transition/reattachment in the leading-edge area and the
flow in the near wake further downstream of the trailing-edge noise source center. The near-wake source is observed to dominate the high-frequency noise radiation.

A weakly compressible flow direct numerical simulation of a controlled-diffusion aerofoil at 8 degree geometrical angle of attack, a chord based Reynolds number of Rec = 150000 and a Mach number of M = 0.25 based on the free-stream velocity relevant to many industrial applications, was conducted to improve the understanding of the impact of the pressure gradient on the development of turbulent structures. The evolution equations for the two invariants Q and R of the velocity gradient tensor have been studied at various locations along the aerofoil chord on its suction side. The shape of the mean evolution of the velocity gradient tensor invariants were found to vary strongly when the flow encounters favorable, zero and adverse pressure gradient and as well for different wall normal locations. The coupling between the pressure-Hessian tensor and the velocity gradient tensor was found to be the major factor that causes these changes and is greatly influenced by the mean pressure gradient condition and the wall normal distance. Striking differences exist from the mean trajectories of this coupling at least in the log-layer and outer-layer subject to different mean pressure gradients. The non-linearity and viscous diffusion effects keep their respective invariant characters regardless of the pressure gradient effects and wall normal locations. The wall and the mean adverse pressure gradient were found both to suppress the vortical stretching features of the flow. These features are of great importance for the development of future turbulence models on wall bounded flows, especially on surfaces with significant curvature such as cambered aerofoils and blades for which significant mean pressure gradients exist.

A 3D compressible direct numerical simulation is .conducted of a controlled-diffusion
airfoil at 8 angle of attack that is embedded in a wind-tunnel flow at an airfoil chord based
Reynolds number of Re = 150, 000 and at a Mach number of M = 0.25. Under these flow
conditions, a short separation bubble is formed at the airfoil leading-edge on the suction side which triggers the laminar to turbulent transition of the flow. The flow remains turbulent and attached until the trailing-edge. On the pressure side, the flow stays laminar until.the trailing-edge where minor vortex shedding originates due to the blunt (round) trailing-edge and then mixes with the turbulent flow from the suction side in the near wake. The DNS domain comprises the near field around the airfoil. The aerodynamic installation effect of the wind tunnel is included by using an appropriate set of inflow boundary profiles, a technique that has been used successfully in previous numerical, incompressible, airfoil studies. The hydrodynamic field is firstly compared with hot-wire and particle image velocimetry measurements to ensure a proper reproduction of the experimental conditions. A comparison with a previous direct numerical simulation using Lattice-Boltzmann method is also presented. The acoustic far-fleld is then computed from the near field solution, using the Ffowcs Williams and Hawkings equations for a porous control surface and a solid surface. Both Ffowcs WilIiams Hawkings surfaces show a good agreement with experimental data for the low to mid frequency range. A high frequency hump is found from the porous Ffowcs Williams Hawkings surface which is possibly caused by a secondary noise source in the near wake.

The broadband noise sources are investigated on an isolated low-speed fan typical of engine cooling systems. RANS simulations have been performed on a single blade passage for several flow rates at the same rotational speed. The flow structures responsible for the different noise contributions are identifi�ed by a systematic analysis of the simulation results. The aeroacoustic noise predictions are based on Amiet's model for rotating sources in free-fi�eld. The contribution of the turbulence-interaction noise and the trailing-edge noise are considered by the appropriate isolated blade response and statistical model of the turbulent sources. The flow parameters of the aeroacoustic response and the turbulent models are extracted from the RANS simulations. The radial evolution of the flow parameters for the different flow rates is analyzed and related to the three dimensional flows in the machine. The acoustic predictions are validated with experimental spectra measured upstream of the fan in a reverberant room. The two considered mechanisms evolve differently with the flow rate. The leading edge sources are dominant at low flow rate up to the design point and the self noise becomes dominant at high flow rate for which the secondary flow structures are limited.

Direct numerical simulations are used to examine the behaviour of wall-pressure fluctuations $p_{w}$ in a flat-plate turbulent boundary layer with large adverse and favourable pressure gradients, involving separation and reattachment. The Reynolds number $Re_{\unicode[STIX]{x1D703}}$ based on momentum thickness is equal to 300, 600 and 900. Particular attention is given to effects of Reynolds number on root-mean-square (r.m.s.) values, frequency/power spectra and instantaneous fields. The possible scaling laws are also examined as compared with the existing direct numerical simulation and experimental data. The r.m.s. value of $p_{w}$ normalized by the local maximum Reynolds shear stress $-\unicode[STIX]{x1D70C}\overline{uv}_{max}$ (Simpson et al. J. Fluid Mech. vol. 177, 1987, pp. 167–186; Na & Moin J. Fluid Mech. vol. 377, 1998 b , pp. 347–373) leads to near plateau (i.e. $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}=2.5\sim 3$ ) in the adverse pressure gradient and separated regions in which the frequency spectra exhibit good collapse at low frequencies. The magnitude of $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}$ is however reduced down to 1.8 near reattachment where good collapse is also obtained with normalization by the local maximum wall-normal Reynolds stress $\unicode[STIX]{x1D70C}\overline{vv}_{max}$ . Near reattachment, $p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{vv}_{max}=1.2$ is attained unambiguously independently of the Reynolds number and pressure gradient. The present magnitude (1.2) is smaller than (1.35) obtained for step-induced separation by Ji & Wang ( J. Fluid Mech. vol. 712, 2012, pp. 471–504). The reason for this difference is intrinsically associated with convective nature of a pressure-induced separation bubble near reattachment where the magnitude of $p_{w\,rms}$ depends essentially on the favourable pressure gradient. The resulting mean flow acceleration leads to delay of the r.m.s. peak after reattachment. Attention is also given to structures of $p_{w}$ . It is shown that large-scale spanwise rollers of low pressure fluctuations are formed above the bubble, whilst changing to large-scale streamwise elongated structures after reattachment. These large-scale structures become more prominent with increasing $Re_{\unicode[STIX]{x1D703}}$ and affect $p_{w}$ significantly.

A more complete understanding of the physical relationships, between wall-pressure and turbulence, is required for modeling flow-induced noise and developing noise reduction strategies. In this study, the wall-pressure fluctuations, induced by low Reynolds numberturbulent boundary layers, are experimentally studied using a high-resolution microphone array. Statistical characteristics obtained using traditional cross-correlation and cross-spectra analyses are complimented with wall-pressure–velocity cross-spectra and wavelet cross-correlations. Wall-pressure–velocity correlations revealed that turbulent activity in the buffer layer contributes at least 40% of the energy to the wall-pressure spectrum at all measured frequencies. As Reynolds number increases, the low-frequency energy shifts from the buffer layer to the logarithmic layer, as expected for regions of uniform streamwise momentum formed by hairpin packets. Conditional cross-spectra suggests that the majority of broadband wall-pressure energy is concentrated within the packets, with the pressure signatures of individual hairpin vortices estimated to decay on average within traveling ten displacement thicknesses, and the packet signature is retained for up to seven boundary layer thicknesses on average.

This new edition of the near-legendary textbook by Schlichting and revised by Gersten presents a comprehensive overview of boundary-layer theory and its application to all areas of fluid mechanics, with particular emphasis on the flow past bodies (e.g. aircraft aerodynamics). The new edition features an updated reference list and over 100 additional changes throughout the book, reflecting the latest advances on the subject.

The statistical properties of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) are presented. The flow is generated via the direct numerical simulation of a TBL on a flat surface with a farfield boundary condition designed to apply the desired pressure gradient. The conditions for self-similarity and appropriate scaling are derived, with the mean profiles, Reynolds stress profiles, and turbulent kinetic energy budgets non-dimensionalised using this scaling. The APG TBL has a momentum thickness based Reynolds number range from to 6000, with a self-similar region spanning a Reynolds number range from to 4800. Within this range the non-dimensional pressure gradient parameter . Two-point correlations of each of the velocity components in the streamwise/wall-normal plane are also presented, which illustrate the statistical imprint of the structures in this plane for the APG TBL.

A turbulent boundary layer (TBL) can be an important source of noise and vibration and its simulation is still an open research challenge. The stochastic pressure distribution associated with turbulence can significantly excite a structure that radiates acoustic power. In such a situation, a good description of the wall pressure field is necessary for an accurate prediction of the vibration and noise propagation. In order to tackle this issue, many TBL models have been developed since the 50s. Among others, the Corcos model has been widely used, especially because of its advantageous mathematical features. However, a major drawback is the small rate of decay for wavenumbers below the coincidence frequency.

Spectral density, magnitude of the normalized longitudinal and lateral cross‐spectral‐density functions, and convection‐velocity ratio as a function of longitudinal separation and frequency of wall‐pressure fluctuations were measured with small flush‐mounted transducers. These measurements were accomplished in both mild adverse and mild favorable pressure gradients in a low‐turbulence subsonic wind tunnel. To establish a basis of comparison, similar measurements were made for the zero pressure gradient, and agreement with published measurements was excellent. The effect of an adverse pressure gradient on the nondimensionalized spectral density was an increase in the low‐frequency content without influencing the high‐frequency portion appreciably; a sharp decrease in the high‐frequency portion was observed for the favorable pressure gradient. At similar nondimensionalized longitudinal separations and frequencies, the convection velocity ratio was higher in the favorable and lower in the adverse pressure gradients. The longitudinal decay of a particular eddy was more rapid in the adverse and slower in the favorable pressure gradients. No differences were found in the lateral cross‐spectral density for the different pressure gradients.

DNS of a turbulent channel flow has been carried out at four Reynolds numbers, 180, 395, 640 and 1020, based on the friction velocity and the channel half width in order to investigate the Reynolds-number dependence on the pressure fluctuations. It is shown that large peaks appear in the streamwise spectra of the wall pressure fluctuations at low wavenumbers for the Reynolds numbers investigated. The origin of the peaks is examined using the pressure splitting method such as the rapid and slow parts. At higher Reynolds number, a closer examination reveals that large-scale patterns of the instantaneous wall pressure fluctuations are essentially associated with large-scale structures of the instantaneous rapid pressure in the outer layer, which causes the noticeable peaks in the wall pressure spectra at low wavenumbers.

Space-time correlation measurements in the roughly isotropic turbulence behind a regular grid spanning a uniform airstream give the simplest Eulerian time correlation if we choose for the upstream probe signal a time delay which just ‘cancels’ the mean flow displacement. The correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency ([similar] wave-number at these small turbulence levels). With plausible scaling of the time separations, a scaling dependent on both wave-number and time, it is possible to effect a good collapse of the correlation functions corresponding to wave-numbers from 0·5 cm−1, the location of the peak in the three-dimensional spectrum, to 10 cm−1, about half the Kolmogorov wave-number. The spectrally local time-scaling factor is a ‘parallel’ combination of the times characterizing (i) gross strain distortion by larger eddies, (ii) wrinkling distortion by smaller eddies, (iii) convection by larger eddies and (iv) gross rotation by larger eddies.

This paper reports the results of an extensive experimental investigation into the mean flow properties of turbulent boundary layers with momentum thickness Reynolds numbers less than 3000. Zero pressure gradient and favorable pressure gradients were studied. The velocity profiles displayed a logarithmic region even at very low Reynolds numbers (as low as R SUB THETA =261). The results were independent of the loading edge shape, and the pin type turbulent simulators performed well. It was found that the shape and Clauser parameters were a little higher than the correlation proposed by Coles and the skin friction coefficient was a little lower. The skin friction coefficient was a little lower. The skin friction coefficient behaviour could be fitted well by a simple power law relationship in both zero and favourable pressure gradients. (A)

Airfoil self-noise is the noise produced by the interaction between an airfoil with its own boundary layers and wake. Self-noise is of concern as it is an important contributor to the overall noise in many applications, e.g. wind turbines, cooling fan blades, or air frames, to name a few. The continued growth of available computing power has made direct numerical simulations (DNS) of compressible flows with application to airfoil noise possible. Challenges associated with such simulations and numerical details of a DNS code that is able to exploit modern high-performance computing systems are presented. Results obtained from DNS of flow over NACA-0012 airfoils at moderate Reynolds number are used to evaluate the accuracy of approximations commonly made in deriving trailing edge noise theories. The data are also used to identify additional noise sources present in airfoil configurations. Finally, DNS are employed to study the noise reducing effect of trailing-edge noise serrations, indicating that the noise reduction is mainly due to a changed scattering mechanisms and not a change in the incidence turbulence field.

In the flow of air past a solid surface, a turbulent boundary layer is produced. The turbulence is ordinarily thought of as velocity fluctuation but in addition pressure, temperature and density fluctuations also occur. The turbulent boundary layer causes a sound field to be set up in the free stream and also induces fluctuating normal loads on the boundary. If the boundary is elastic, its motion will set the air on each side into motion. Thus, additional sound radiation is to be expected. In an initial investigation of this problem some of the properties of the wall pressurefluctuations in a turbulent boundary layer have been measured. The equipment used in the investigation includes: a specially designed low noise and turbulence level wind tunnel, a small barium titanate microphone for frequencies up to 50 kc and the necessary electronic equipment. The quantities measured were the root‐mean‐square pressure at the wall and the power spectrum of the wall pressure. It was found that the root‐mean‐square wall pressure was a constant portion (0.0035) of the free stream dynamic pressure for 0.2 <M <1 and 106 <R <13 × 106. [Supported by the National Advisory Committee for Aeronautics.]

When a turbulent boundary layer is produced by air flow past a solid surface, the turbulence in the boundary layer can generate a sound field in the free stream and will also induce fluctuating loads on the solid surface. If the surface is flexible, this motion will generate an additional sound field on both sides of the surface.
In an initial investigation of the latter form of sound generation, suitable equipment has been developed to measure the fluctuating wall pressure in the turbulent boundary layer. The equipment includes a specially designed low noise and turbulence level wind tunnel and a small barium titanate transducer and preamplifier combination for frequencies up to 50 kc. The transducer and preamplifier may be useful for other applications.
Using this equipment, some of the properties of the wall pressure fluctuations in a turbulent boundary layer have been measured. It was found that the spectrum of the wall pressure fluctuations extends to 50 kc and that the root-mean-square wall pressure was a constant portion (0.0035) of the free stream dynamic pressure for 0.2 <M <0.8 and 1.5×106<Re<20×105. A few typical spectra are given for different values of the Reynolds number and Math number.

Broadband noise produced by the trailing-edge of a controlled diffusion (CD) airfoil is directly simulated using a Lattice-Boltzmann method (PowerFlow) that resolves both the aerodynamic and acoustic field around the airfoil. The proper DNS resolution is first achieved in the vicinity of the airfoil on a quasi-2D slice of the mock-up. It is then extended to a 3D slice with a span of 12 % chord length. Two numerical setups of the anechoic open- jet facility where both aerodynamic and acoustic data have been collected are investigated to capture the installation effects: in a first numerical setup (called free), the CD airfoil is set in an uniform flow, while in the second setup (lips) the real jet nozzle geometry is considered. While in the free-field configuration the boundary layer rapidly detaches on the suction side, in the lips the jet shear layers modify the pressure load on the airfoil and the boundary layer keeps attached in the configuration with nozzle. In both setups a laminar recirculation bubble is captured on the suction side near the leading edge which triggered the development of the boundary layers along the suction side. The wall-pressure and noise spectra for the free configuration are spread over a large frequency band and agree with similar measurements at higher angle of attack for which the flow is detached. The spectra for the lips configuration better agree with the experimental data and has been selected for the 3D simulation. The vortex stretching along the span-wise direction that was missing in the previous investigated set-ups, allows to finely capture the turbulence length scales and accurately reproduce the experimental measurements. Both the boundary layer profile and the wall pressure spectra near the trailing edge are nicely and accurately captured. The predicted far-field sound pressure levels also provide satisfactory agreement with noise measurements in the anechoic wind tunnel. © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Accurate measurements of the turbulent wall pressure have been difficult to achieve due to signal contamination at low frequencies by background facility noise and/or attenuation at high frequencies due to sensor averaging effects. The current study utilizes a new noise cancellation scheme based on an optimal filtering technique to capture the noise-canceled time series. Furthermore, to address the high-frequency attenuation problem, a number of pinholes are utilized along with high-sensitivity microphones to obtain measurements of the wall pressure at a resolution down to d+ = duτ/v ≅ 2. The results show that the maximum allowable nondimensional sensing diameter to avoid spectral attenuation for frequencies up to f+ = fv/uτ2 = 1 is in the range of 12.0 ≤ d+ < 18.0. Additionally, it is shown that the wall-pressure rms level when scaled on the wall-shear stress seems to increase slowly with increasing Reynolds number, apparently due to the influence of large-scale structures in the outer part of the boundary layer. On the other hand, it is demonstrated that, even though √p′2/q0 appears to become invariant with increasing Reynolds number, p′ is not generated by structures whose velocity scale is U0.

A nonreflecting zonal boundary condition was presented for direct numerical simulation of aerodynamic sound. The method is based on commonly used characteristic boundary conditions and, therefore, only requires marginal modifications to existing codes. It was shown that the proposed zonal boundary condition succeeds in significantly reducing spurious pressure oscillations that are produced when vortical structures in compressible flow cross the far-field boundaries. The aim of these simulations was to evaluate the acoustic scattering of Tollmien-Schlichting (T-S) waves convecting over the trailing edge(TE).

Space-time correlations and frequency spectra of wall-pressure fluctuations, obtained from direct numerical simulation, are examined to reveal the effects of pressure gradient and separation on the characteristics of wall-pressure fluctuations. In the attached boundary layer subjected to adverse pressure gradient, contours of constant two-point spatial correlation of wall-pressure fluctuations are more elongated in the spanwise direction. Convection velocities of wall-pressure fluctuations as a function of spatial and temporal separations are reduced by the adverse pressure gradient. In the separated turbulent boundary layer, wall-pressure fluctuations are reduced inside the separation bubble, and enhanced downstream of the reattachment region where maximum Reynolds stresses occur. Inside the separation bubble, the frequency spectra of wall-pressure fluctuations normalized by the local maximum Reynolds shear stress correlate well compared to those normalized by free-stream dynamic pressure, indicating that local Reynolds shear stress has more direct influence on the wall-pressure spectra. Contour plots of two-point correlation of wall-pressure fluctuations are highly elongated in the spanwise direction inside the separation bubble, implying the presence of large two-dimensional roller-type structures. The convection velocity determined from the space-time correlation of wall-pressure fluctuations is as low as 0.33U 0 (U 0 is the maximum inlet velocity) in the separated zone, and increases downstream of reattachment.

The present work reviews papers which have significantly contributed to
the understanding of pressure fluctuations caused by a turbulent
boundary layer. Theoretical models based on experimental measurements
have been mostly used to provide approximate results for the statistics
of the turbulent pressure field. The pressure field in a turbulent flow
is produced by a summation of contributions from the turbulent velocity
fluctuations. Attention is restricted to incompressible flows in which
pressure fluctuations are related to the velocity fluctuations through
Poisson's equation, obtained from the divergence of the momentum
equation. The review considers convection and decay of wall pressure
fluctuations, the effect of transducer size on measurements of wall
pressure fluctuations, measurements of the power spectrum and rms wall
pressure fluctuation, the spatial scale of wall pressure fluctuations,
and the effect of the pressure gradient and of roughness on wall
pressure fluctuations.

Wall pressure data acquired during flight tests at several flight conditions are analysed and the correlation and coherence lengths of the data reported. It is found that the correlation and coherence lengths are influenced by the origin of the structure producing the pressure and the frequency bandwidth over which the analyses are performed. It is shown how the frequency bandwidth biases the correlation length and how the convection of the pressure field might reduce the coherence measured between sensors. A convected form of the cross correlation and cross spectrum is introduced to compensate for the effects of convection. Coherence lengths measured in the streamwise direction appear much longer than expected. Coherent structures detected using the convected cross correlation do not exhibit an exponential coherent power decay.

Direct and large-eddy simulations (DNS and LES) of spatially developing high-Reynolds number turbulent boundary layers (Re theta up to 4300) under zero pressure gradient are studied. The inflow of the computational domain and the tripping of the boundary layer is located at low